Decomposition of self-similar stable mixed moving averages
A standard Borel space (SBS) is a measurable space measurable isomorphic to a Borel subset of a complete separable metric space. When a SBS is equipped with a \(\sigma\)-finite measure \(\mu\), it is called standard Lebesgue space (SLS). Let \((X, \mathcal X, \mu)\) be a SLS. A family \(\{\varphi_t\}_{t\in\mathbb R}\) of measurable maps from \(X\) onto \(X\) form an additive flow if \(\varphi_0(x)=x\) and \(\varphi_{t_1}(\varphi_{t_2}(x))=\varphi_{t_1+t_2}(x)\). The flow \(\{\varphi_t\}\) is nonsingular if \(\mu(\varphi_t^{-1}(B))=0\) iff \(\mu(B)=0\) for every \(t\in\mathbb R\) and \(B\in\mathcal X\). A measurable map \(a_t(x):\mathbb R\times X\mapsto \{-1,1\}\) is a cocycle if \(a_{t_1+t_2}(x)=a_{t_2}(x)a_{t_1}(\varphi_{t_2}(x))\). The authors investigate a symmetric \(\alpha\)-stable (S\(\alpha\)S) mixed moving average process \(X_{\alpha}\) with stationary increments defined by \[ X_{\alpha}(t)=\int_X\int_{\mathbb R} (G(x, t+u) -G(x,u))M_{\alpha}(dx,du) \] where \(G\) is a measurable function and \(M_{\alpha}\) is a S\(\alpha\)S random measure with the control measure \(m_{\alpha}(dx,du)=\mu(dx)du\). It is proved that a self-similar process \(X_{\alpha}\) is determined by a nonsingular flow, a related cocycle and a semi-additive functional. A unique decomposition into two independent components \(X_{\alpha}\overset\text{d}= X_{\alpha}^D +X_{\alpha}^C\) is established, where \(X_{\alpha}^D\) is determined by a nonsingular dissipative flow and \(X_{\alpha}^C\) by a nonsingular conservative flow.
stable self-similar process with stationary increments
scientific article; zbMATH DE number 1836949